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If y = log_a x, x>0, then dy/dx =

(1) 1/x (2) 1/ax (3) (1/x)logx a (4) (1/x)loge a Solution: y = loga x =log x/log a dy/dx =... View Article

(d/dx)(1/x⁴ sec x) =

(1) [4 cos x – x sin x]/x5 (2) -[x sin x + 4 cos x]/x5 (3) [4 cos x... View Article

If y = x + 1/x, then

(1) x2(dy/dx) + xy = 0 (2) x2(dy/dx) + xy + 2 = 0 (3) x2(dy/dx) – xy + 2... View Article

[latex]\frac{d}{dx}\left ( \sqrt{x}+\frac{1}{\sqrt{x}} \right )^{2}=[/latex]

(1) 1 – 1/x2 (2) 1 + 1/x2 (3) 1 – 1/2x (4) none of these Solution: Let y =... View Article

If u = sin^-1 ((x²+y²)/(x+y)) , then x ∂u/∂x + y ∂u/∂y is equal to

(1) sin u (2) tan u (3) cos u (4) cot u Solution: u = sin-1 ((x2+y2)/(x+y)) sin u =... View Article

(d/dx)log tan x =

(1) 2 sec 2x (2) 2 cosec 2x (3) sec 2x (4) cosec 2x Solution: Let y = log tan... View Article

If z = log (tan x + tan y), then sin 2x(∂z/∂x) + sin 2y(∂z/∂y) is equal to

(1) 1 (2) 2 (3) 3 (4) 4 Solution: Given z = log (tan x + tan y) Take antilog... View Article

If z = tan (y + ax) – √(y-ax), then z_xx-a²z_yy is equal to

(1) 0 (2) 2 (3) zx+zy (4) zxzy Solution: Given z = tan (y + ax) – √(y-ax) Differentiate partially... View Article

Derivative of sec^-1 (1/1-2x²) with respect to sin^-1 (3x – 4x³) is

(1) 1/4 (2) 3/2 (3) 2/3 (4) 1 Solution: Let u = sec-1(1/1-2x2) Put x = sin θ θ =... View Article

If [latex]y=\tan^{-1}\left ( \frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}} \right )[/latex] , then dy/dx is equal to

(1) x2/√(1-x4) (2) x2/√(1+x4) (3) x/√(1+x4) (4) x/√(1-x4) Solution: Given y = tan-1 (√(1+x2) – √(1-x2) )/(√(1+x2) + √(1-x2))) Put... View Article

If x = cos θ, y = sin 5θ, then (1-x²)d²y/dx² – xdy/dx is equal to

(1) -5y (2) 5y (3) 25y (4) -25y Solution: Given x = cos θ y = sin 5θ dx/dθ =... View Article

(1) (d2y/dx2)-1 (2) -(d2y/dx2)-1(dy/dx)-3 (3) -(d2y/dx2)(dy/dx)-3 (4) none of these Solution: dx/dy = 1/(dy/dx) \(\begin{array}{l}\frac{d^{2}x}{dy^{2}}= \frac{d}{dy}\left ( \frac{dx}{dy} \right )\end{array}... View Article

If y = sin (log_ex) then x² d²y/dx² + xdy/dx is equal to

(1) sin (logex) (2) cos (logex) (3) y2 (4) -y Solution: Given y = sin (logex) Differentiate w.r.t.x dy/dx =... View Article

If x = log_e t, t > 0 and y+1 = t². Then d²y/dx² is equal to

(1) 4e2x (2) (-½)e-4x (3) (-¾)e-5x (4) 4ex Solution: Given x = loge t So ex = t Given y+1... View Article

If [latex]f(x)=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+…x^{n}[/latex], then f’’(1) is equal to

(1) n(n-1)2n-1 (2) (n-1)2n-1 (3) n(n-1)2n-2 (4) n(n-1)2n Solution: f(x) = 1 + nx + n(n-1)x2/2! + n(n-1)(n-2)x3/3! +….+xn =... View Article

[latex]y=e^{a\sin^{-1}x}[/latex], then [latex](1-x^{2})y_{n+2}-(2n+1)xy_{n+1}[/latex] is equal to

1) – (n2 + a2)yn 2) (n2 – a2)yn 3) (n2 + a2)yn 4) -(n2 – a2)yn Solution: Given \(\begin{array}{l}y=e^{a\sin^{-1}x}\end{array}... View Article

If f (x) = tan^–1 x, Then, f ‘ (x) + f ”(x) = 0, where x is equal to

1) 0 2) 1 3) i 4) -i Solution: f (x) = tan–1 x Differentiate w.r.t.x f’(x) = 1/(1+x2) f’’(x)... View Article

If y = sec (tan^–1 x), then dy/dx at x = 1 is equal to

1) 1/√2 2) 1/2 3) 1 4) √2 Solution: y = sec (tan–1 x) dy/dx = sec (tan–1 x) tan... View Article

If x = a cos³ θ and y = a sin³ θ, 1 + (dy/dx)² is equal to

1) tan θ 2) tan2 θ 3) 1 4) sec2 θ Solution: Given x = a cos3 θ dx/dθ =... View Article

If x = sin t and y = tan t, then dy/dx is equal to

(1) cos3 t (2) 1/cos3 t (3) 1/cos2 t (4) 1/sin2 t Solution: Given x = sin t y =... View Article

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